Monochromatic balanced components, matchings, and paths in multicolored complete bipartite graphs

TL;DR

This paper investigates the existence of large, balanced monochromatic components in multicolored complete bipartite graphs, proving positive results for up to three colors and constructing counterexamples for four or more colors.

Contribution

It provides a short proof for the existence of balanced monochromatic components in up to three colors and shows that such balanced components do not necessarily exist for four or more colors, also determining the bipartite Ramsey number for paths.

Findings

For r ≤ 3, every r-coloring of K_{m,n} has a balanced monochromatic component meeting each side in at least 1/r of its vertices.

For r ≥ 4, there are colorings where the largest balanced monochromatic component is smaller than n/r, specifically n/5 for r=4.

The paper determines the bipartite Ramsey number for P_4 for all r.

Abstract

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that meets both sides in at least $n/r$ vertices? Over forty years ago, Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Buci\'c, Letzter and Sudakov proved that every $3$-colored $K_{n,n}$ contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size $\lceil n/3 \rceil$. So the answer is strongly "yes" for $1\leq r\leq 3$. We provide a short proof of (a non-symmetric version of) the original question for $1\leq r\leq 3$; that is, every $r$-coloring of $K_{m,n}$ has a monochromatic component that meets each side in a $1/r$ proportion of its part size. Then, somewhat surprisingly, we show that the answer to the question is "no" for all $r\ge 4$. For instance, there are $4$-colorings of $K_{n,n}$ where the largest balanced monochromatic component has $n/5$ vertices in both partite classes (instead of $n/4$). Our constructions are based on lower bounds for the $r$-color bipartite Ramsey number of $P_4$, denoted $f(r)$, which is the smallest integer $\ell$ such that in every $r$-coloring of the edges of $K_{\ell,\ell}$ there is a monochromatic path on four vertices. Furthermore, combined with earlier results, we determine $f(r)$ for every value of $r$.